Appendix D
Glossary of Common Terms and Abbreviations in Quantum Chemistry

The literature of quantum chemistry, covering both molecular orbital
theory and density functional theory, is cluttered with abbreviations,
acronyms, and jargon. Some of the more common terminology is explained
in this glossary.

An extensive list of acronyms has been published recently (1),
and a major reference work is currently in preparation. The list below
is adapted from one that was developed for a brief course within the Physical
and Chemical Properties Division at NIST (2).

Glossary

The asterisk indicates that a set of polarizing d-functions (6D) is
included to supplement the 3-21G basis, but only on second-row and heavier
atoms (beyond neon). Also denoted 3-21G(*) or 3-21G(d).

5D

Indicates that five functions are used in each d-set.

6D

Indicates that six (cartesian) functions are used in each d-set. This
includes the s-like combination (x2 + y2 + z2).

6-31G

A VDZ basis set from the Pople school. Very popular,
often used with a set of heavy-atom polarization functions (see 6-31G*).
The "6" indicates that each core basis function is built using six primitives.
The "3" indicates that the inner valence basis functions are each built
using three primitives. The "1" indicates that the outer valence basis
functions are each built using a single uncontracted primitive. The "G"
stands for "Gaussian", indicating the type of primitive function. (Recently-developed
basis sets don't include the "G" or its equivalent since they are essentially
all based upon Gaussian functions.)

6-311G

A popular VTZ basis set similar to the small 6-31G
set. Usually supplemented with polarization functions. Built like 6-31G
but with a third layer of valence functions composed of a single, uncontracted
primitive set. Some workers consider this basis to be less flexible than
a "real" triple-zeta basis.

6-311G*

The asterisk indicates that a set of polarization d-functions (5D)
has been added to heavy atoms to supplement the 6-311G basis; also denoted
6-311G(d).

6-311G**

The second asterisk indicates that a set of polarization p-functions
has been added to hydrogen; also denoted 6-311G(d,p).

6-311+G(3df,2p)

In addition to the 6-311G basis, the "+" indicates that diffuse s-
and p-functions are added to heavy atoms, the "3df" indicates that three
sets of polarization d-functions and one set of polarization f-functions
are added to heavy atoms, and the "2p" indicates that two sets of polarization
p-functions are added to hydrogen.

6-31G*

A polarized VDZ basis set from the Pople school.
Maybe the most popular basis set in use today. The single asterisk indicates
that a set of polarizing d-functions (6D) is included
on "heavy" atoms (beyond helium). Also denoted 6-31G(d).

6-31G**

A polarized VDZ basis set from the Pople school.
A set of polarizing d-functions (6D) is included on "heavy"
atoms and a set of p-functions on hydrogen. Also denoted 6-31G(d,p).

6-31+G*

Augmented 6-31G* basis; the single "+" indicates that a set of diffuse
s-functions and a set of diffuse p-functions has been added to each heavy
atom. Also denoted 6-31+G(d).

6-31++G*

Augmented 6-31+G* basis; the second "+" indicates that a set of diffuse
s-functions has been added to each hydrogen atom. Also denoted 6-31++G(d).

Atoms-in-molecules. An analysis method based upon the shape of the
total electron density; used to define bonds, atoms, etc. (5).
Atomic charges computed using this theory are often quite different from
those from other analyses (e.g., from Mulliken populations). Such charges
are probably the most justifiable theoretically, but meet some resistance
because the values obtained may be quite different from those from older
theories (6).

Atomic natural orbital. Very large basis sets derived from correlated
atomic calculations. More expensive to use than the corresponding correlation
consistent basis sets (e.g., cc-pVDZ) but not often
significantly more accurate.

The set of mathematical functions (basis functions) used to describe
the molecular orbitals. Gaussian functions predominate heavily, but
occasional papers use the old "Slater" orbitals or functions, which are
exponentials.

beta electron

An electron with spin down.

B3LYP

DFT using Becke exchange functional and Lee-Yang-Parr correlation functional,
as well as Hartree-Fock exchange.
A hybrid method; the parameters were
optimized for thermochemistry, but using a different functional and numerical
(basis set-free) code (7,8). Very popular.

BAC-MP4

Bond-additivity-corrected MP4. A method in which an MP4 energy is corrected
using empirical parameters that depend upon the atoms in the molecule,
bond distances, and nearest neighbors. Method developed by Carl Melius
(Sandia); not well-described in the open literature.

Basis-set superposition error. An insidious artifact traceable to the
fact that one can seldom afford to use a really big basis set. It causes
an extra decrease in energy (i.e. more negative energy, greater stability)
when two systems (atoms or molecules) are brought together. The energy
of one fragment is lowered because its orbitals can use the basis functions
on the other fragment, even if the actual electrons and nuclei on the other
fragment are not included in the calculation. With a complete basis set,
there is no BSSE because the other fragment's basis functions are superfluous.
BSSE is usually ignored in thermochemical calculations, except for studies
of weak, non-bonded interactions. See "counterpoise."

The positions in space of the atoms in a molecule listed as triples
(x, y, z).

CASPT2

Complete active space, second-order perturbation theory. This is one
formulation of MP2 theory using a CASSCF
reference instead of a HF reference. A high-level multireference
theory.

CASSCF

Complete active space self-consistent field. A type of MCSCF
calculation
in which the configurations are chosen to be all those obtainable (i.e.,
full CI ) using a specified number of electrons and a
specified set of orbitals. The set of orbitals is called the "active space,"
and the specified electrons are called "active." In many cases, it requires
experience and skill to select the active space correctly.

CBS

Complete basis set. Indicates that some method of basis set extrapolation
was applied in an attempt to determine the result that would have been
obtained using an infinitely large basis set.
The two major extrapolation methods are (1) repeating the calculation
with increasingly large basis sets and making an empirical extrapolation,
and (2) using analytical formulas that are correct to second-order.
See the chapters by Martin and by Petersson in this book.

cc-pVDZ

Correlation-consistent polarized valence double-zeta basis set. The
smallest in a series of "correlation consistent" basis sets developed by
Dunning and coworkers for high-level calculations. It has been observed
that properties computed using successively larger basis sets of this series
appear to converge exponentially, presumably to the corresponding CBS
values.

Coupled-cluster, singles and doubles with approximate triples. (See
CCD.) Triples contributions are determined perturbatively. CCSD(T) is the
cheapest of the usual approximations to full CCSDT, but appears to be the
best. The most popular high-level (i.e., lots of electron correlation)
method. Size-consistent. Very expensive.

Contracted gaussian-type orbital. The usual basis function; it's a
linear combination of gaussian functions with the linear coefficients fixed,
then multiplied by an angular function. So one set of p-functions contains
three cGTO's (px, py, and pz), i.e, three
basis functions. See "primitives."

CI

Configuration interaction. A theory of electron correlation.
A large set of Hartree-Fock-type configurations
(Slater determinants) is used as a many-electron basis set.
The coefficient of each configuration is determined
variationally so as to minimize the total energy of this wavefunction.
Recovers the "dynamic" electron correlation important in bonding.
Reliability and expense depend upon the size of the CI, e.g., CISD was popular before
coupled-cluster methods caught on. Ordinary, truncated CI (CIS, CISD, etc.)
is not size-consistent, so determining bond energies requires
"supermolecule"
calculations.

CID

Configuration interaction, doubles. A CI that only
includes those determinants that correspond to double excitations from
the reference (which is usually Hartree-Fock).

CIS

Configuration interaction, singles. (See CID.) The simplest method
for calculating electronically excited states; limited to singly-excited
states. Contains no electron correlation and has no effect on the ground
state (Hartree-Fock) energy.

CISD

Configuration interaction, singles and doubles. A CI
that
only includes those determinants that correspond to single or double excitations
from the reference (which is usually Hartree-Fock).
Declining popularity.

Refers to the particular choice of scheme for generating the linear
combinations of gaussian functions that constitute a contracted basis set.
(See cGTO.) A "generally-contracted" basis set is one
in which each primitive is used in many basis functions. A "segmented"
basis set, in contrast, is one in which each primitive is used in only
one (or maybe two) contracted function.

correlation effects

The effect upon the quantity of interest attributable to the inclusion
of dynamic electron correlation.

correlation energy

The difference between the Hartree-Fock
energy and the FCI energy for a given basis set. Most
of this energy is attributable to the correlation among the positions of
electrons of opposite spin, caused by their coulombic repulsion.
See dynamic and non-dynamic correlation.

counterpoise

The most common, but still controversial, correction for BSSE.
The BSSE is approximated as the energy difference between (1) an isolated
fragment and (2) the fragment accompanied by the basis functions, but not
the atoms, of its companion fragment(s).

Density-functional theory. Ab initio method not based upon a wavefunction.
Instead, the energy is computed as a functional of the electron density.
Sometimes called Kohn-Sham theory. The correct functional has not yet been
found, but many approximations are in use.

diffuse

Diffuse basis functions are typically of low angular momentum (unlike
polarization functions) but with much smaller exponents, so that they spread
more thinly over space. Usually essential for calculations involving negative
ions or Rydberg states.

dihedral angle

The angle between two intersecting planes. In a molecule with atoms
A-B-C-D, the dihedral angle (A-B-C-D) is the angle between the planes defined
by (A, B, C) and by (B, C, D). By convention, the angle is positive for
a right-handed rotation from the first plane to the second, i.e., for a
right-handed twist along the sequence A-B-C-D.

DIIS

Direct inversion in the iterative subspace. An extrapolation procedure
used to accelerate the convergence of SCF calculations.

Double-zeta. A basis set for which there are twice as many basis functions
as are minimally necessary (see "MBS"). "Zeta" (Greek letter
) is the usual name for the exponent that characterizes a Gaussian function.

DZP

Double-zeta with polarization. DZ with polarization
basis functions added. A polarization set generally has an angular momentum
one unit higher than the highest valence function. So a polarization set
on carbon is a set of d-functions.

ECP

Effective core potential. The core electrons have been replaced by
an effective potential. Saves computational expense. May sacrifice some
accuracy, but can include some relativistic effects for heavy elements
(see RECP).

ESP

Electrostatic potential. The electrical potential due to the nuclei
and electrons in the molecule, as experienced by a test charge.

exchange energy

Also called "exchange correlation energy." The energy associated with
the correlation among the positions of electrons of like spin. This is
included in Hartree-Fock calculations.

expensive

Requiring large resources: cpu time, memory, and/or disk space.

FCI

Full configuration interaction. A CI that includes
all possible determinants. FCI is the best wavefunction (and provides the
lowest variational energy) obtainable using a given basis set. Almost never
affordable.

frozen

Indicates that some orbitals were not included in the treatment. Usually
used as "frozen-core," to indicate that the core orbitals were left uncorrelated
in a correlated calculation. Sometimes (esp. in some DFT
programs) it means that core orbitals were fixed as taken from calculations
on isolated atoms.

G1

Gaussian-1. A composite method for computing thermochemistry, involving
extrapolations and a few parameters (9). Effectively superseded
by G2 and G2(MP2).

Gaussian-2. A composite method for computing thermochemistry, involving
extrapolations and a few parameters (10). Very popular
but quite expensive; practical for up to seven
"heavy" atoms (i.e., non-hydrogen) on a Cray supercomputer.

G2(MP2)

Gaussian-2, second-order variant. A composite method for computing
thermochemistry, involving extrapolations and a few parameters (11).
Less-expensive alternative to G2, of comparable
accuracy.

GAMESS

General Atomic and Molecular Electronic Structure System. A free ab
initio software package that emphasizes multi-reference calculations (12).
There is also a British program with the same name, distinguished as GAMESS-UK
(vs. GAMESS-US).

Gaussian

A popular package in the ab initio software industry is by Gaussian Inc
(13).
The name refers to the common use of gaussian functions as basis functions in quantum chemistry.

gaw

Group additivity method as implemented in the
NIST Chemistry Webbook.
See http://webbook.nist.gov/chemistry/grp-add/
for a group additivity method that predicts enthalpies of formation and entropies.
Group additivity methods assume that a molecule can be broken down into separate parts (groups),
each of which contributes additively to the total enthalpy or entropy.
For example acetone is composed of two methyl groups (each adds -42.7 kJ/mol)
and a carbonyl group (which adds -131 kJ/mol) for an estimated enthalpy of formation of -216.4 kJ/mol.

A basis function that is not accompanied by an atomic nucleus, usually
for counterpoise corrections for BSSE.

GIAO

Gauge-independent atomic orbitals. A method specially designed for
calculation of NMR shifts. Currently used with methods such as HF,
MP2,
or DFT. Research codes can handle
CCSD.

GTO

Gaussian-type orbital. Basis function consisting of a Gaussian function,
i.e., exp(-r2), multiplied by an angular function. If the angular
function is "cartesian", there are six d-functions, ten f-functions, etc.
(6d, 10f). If the angular function is spherical, there will the usual number
of functions (5d, 7f).

GVB

Generalized valence bond. A limited type of MCSCF,
in which excitations are taken within an electron pair but not between
orbitals in different pairs. Dissociation-consistent. If restricted to
doubles, is called "perfect pairing" (GVB-PP). If includes both singles
and doubles, is called "restricted configuration interaction" (GVB-RCI).

Simplest and least expensive ab initio wavefunction. Involves only
a single Slater determinant (a single electron configuration). Orbitals
that contain electrons are "occupied," those that are vacant are called
"virtual."

HF

Hartree-Fock. In the Gaussian programs,
HF denotes RHF for closed-shell molecules and UHF
for open-shell.

Highest occupied molecular orbital. The energy of this orbital approximates
the ionization energy of the molecule (Koopmans' theorem).

HOMO-1

The second-highest occupied molecular orbital.

instability

A wavefunction is expressed as a long list of parameters (basis-set
expansion) that are adjusted to minimize the total energy. Sometimes the
global minimum is not obtained; the local minimum that is obtained may
be unstable with respect to various perturbations or liberalization of
constraints. Such a wavefunction is said to be unstable. One of the more
common instabilities is an RHF to UHF instability, which indicates that the
UHF solution (different alpha and beta orbitals) is of
lower energy than the RHF solution
(identical alpha and beta orbitals) for a closed-shell system.
This may be encountered, for example, when a bond is stretched.

Intrinsic reaction coordinate. An optimized reaction path that is followed
downhill, starting from a transition state, to approximate the course (mechanism)
of an elementary reaction step.

isodesmic

Refers to a chemical reaction that conserves types of chemical bond.
Due to better cancellation of systematic errors, energy changes computed
using such reactions are expected to be more accurate than those computed
using reactions that do not conserve bond types. Example: CH3CH2F
+ CH4 = CH3CH3 + CH3F for computing
the C-F bond strength in fluoroethane.

isogyric

Refers to a chemical reaction that conserves net spin. Due to better
cancellation of systematic errors, energy changes computed using such reactions
are expected to be more accurate than those computed using reactions that
do not conserve spin. Example: CH3CH2F + H = CH3CH2
+ HF for computing the C-F bond strength in fluoroethane.

In DFT, a functional that depends only upon the
value of the density, f[rho]. This is the simplest and least expensive
type of functional.

LSDA

Local spin-density approximation. A DFT method involving
only local functionals (i.e., no dependence upon the gradient of the electron
density). In the Gaussian programs "LSDA" is equivalent to the "SVWN" keyword.

LST

Linear synchronous transit. An interpolative method used to guess a
transition state structure given the structures of the products and of
the reactants.

LUMO

Lowest unoccupied molecular orbital. The energy of this orbital is
sometimes used to approximate the electron affinity of the molecule, but
this usually works badly.

Minimal basis set. Only enough basis functions are supplied to put
all the electrons somewhere; the number of basis functions is equal to
the number of orbitals. The most common of these is "STO-3G".
Qualitative results at best.

Multi-configuration self-consistent field. More than one configuration
(Hartree-Fock-type determinant) is used to describe the wavefunction. Both
the coefficients of the configurations and the orbital coefficients are
optimized. This is a limited type of CI (configuration
interaction), with the added feature of orbital optimization. See CASSCF.

Multi-reference configuration interaction. CI using
more that one reference determinant, instead of the usual single
Hartree-Fock
reference. Among multi-reference theories, MR-CISD (singles and doubles
CI) is popular and high-level (but not dissociation consistent).

Mulliken population

A procedure for assigning net atomic charges within a molecule. It
includes an arbitrary choice involving overlap populations, and more seriously
is very sensitive (values varying by more than 100%) to the choice of basis
set. Still used mostly for convenience, since it has no cost and is included
in all ab initio program packages. Superseded by NPA
and AIM methods of population analysis.

The natural orbitals are those for which the first-order density matrix
is diagonal; each will contain some non-integer number of electrons between
0 and 2. Usually discussed in the context of a correlated calculation.
RHF
calculations give molecular orbitals that are also natural orbitals. The
NOs are the orbitals for which the CI expansion converges
fastest.

nondynamic correlation

Also called "static" correlation. The part of the correlation that
is ascribed to the "multireference" nature of the problem at hand, i.e.,
to the qualitative failure of Hartree-Fock theory to describe the system.
The best-known stable molecule with important nondynamic correlation is
singlet methylene, CH2 (Ń 1A1),
for which two configurations are important: (a1)2(b1)0
and (a1)0(b1)2. In many cases, the distinction between
nondynamic and dynamic correlation is rather arbitrary. When nondynamic
correlation is important, single-reference theories may be unreliable.

nonlocal

In DFT, indicates that a functional of the density
gradient (i.e., f[grad rho]) is included in addition to a local functional.
The most popular NL exchange functional is that by Becke. Popular NL correlation
functionals are those by Lee/Yang/Parr and by Perdew. A functional that
includes nonlocal terms is sometimes called "gradient-corrected" or a "GGA,"
which stands for "generalized-gradient approximation."

NPA

Natural population analysis. Considered better than Mulliken populations
for assigning atomic charges; results are fairly independent of the basis
set. Theory based upon chemical concepts of bonds, lone pairs, etc.

orbital

Usually an eigenfunction of a one-electron hamiltonian, e.g., from
Hartree-Fock
theory. A spin orbital has an explicit spin and a spatial orbital does
not. Orbitals are probably the most useful concept from quantum chemistry:
one can think of an atom or molecule as having a set of orbitals that are
filled with electrons (occupied) or vacant (unoccupied or "virtual").

PES

Potential energy surface. The 3N-6 (or 3N-5, for linear molecules)
dimensional function that indicates how the molecule's energy depends upon
its geometry. (Not to be confused with experimental photoelectron spectroscopy.)

Spin-projected MP2 energy. Analog of PUHF,
but for UMP2 energy instead of UHF
energy. Likewise, the PMP3 and PMP4 energies are the UMP3 and UMP4 analogs.

polarized

A "polarized" basis set includes functions that are of higher angular
momentum than is minimally required. For example, carbon atoms have 1s,
2s, and 2p orbitals, so a polarized basis set would also include at least
a set of d-functions. The added functions are often called "polarization
functions." Polarization functions help to account for the fact that atoms
within
molecules are not spherical.

primitives

Also called "primitive functions." The individual gaussian functions
that are summed to produce a contracted basis function (cGTO).
So a set of p-functions is three basis functions, but may be many primitives
(3n, where there are n primitives in the cGTO).

Spin-projected UHF energy. An approximation intended
to provide the energy that would result from a UHF calculation
if it did not suffer spin-contamination. The PUHF energy is usually lower
than the UHF energy because the contributions of higher-multiplicity
states, which usually have high energies, have been (approximately) subtracted.

QCISD with a correction for triples. This method
is at the core of the popular G2 theory. Usually gives
similar results as CCSD(T), of which it is a truncation.

RECP

Relativistic effective core potential. The core electrons have been
replaced by an effective potential that is based upon relativistic quantum
calculations of the free atoms. Saves cost because of fewer explicit electrons
and also includes some relativistic effects, especially the contraction
of core s- and p-orbitals.

redundant internal coordinates

Internal coordinates that overdetermine the
molecular geometry, i.e., are more numerous than 3N-6 for non-linear molecules
or 3N-5 for linear molecules.

reference

As in "single-reference" or "multi-reference," refers to the number
of configurations (or really Slater determinants) in the 0th-order description
of the wavefunction. Most methods that don't begin with "MR," "MC," or
"CAS" are single-reference methods.

RHF

Spin-restricted Hartree-Fock. Closed-shell
singlet with two electrons in each occupied orbital.

ROHF

Spin-restricted open-shell Hartree-Fock.
For open-shell molecules. Except for the odd electron(s), there are two
electrons in each occupied orbital. (See UHF.)

SCF

Self-consistent field. The orbitals (i.e., the coefficients of the
atomic basis functions in each molecular orbital) are adjusted until they
are optimal in the mean electric field that they imply. Implicit for
Hartree-Fock
calculations. Sometimes the term "SCF" is used interchangeably with "HF,"
but it also applies to most DFT calculations and to
all
MCSCF calculations.

SCRF

Self-consistent reaction field. A continuum method for treating solvation.
The simplest formulation involves placing the molecule in a spherical hole
in a polarizable (dielectric) continuum. The molecule polarizes the solvent,
which in turn affects the electron distribution in the molecule; this is
iterated to self-consistency.

scaling

Multiplying calculated results by an empirical fudge factor in the
hope of getting a more accurate prediction. Very often done for vibrational
frequencies computed at the HF/6-31G* level, for which the accepted scaling
factor is 0.893 (17).

An approximate version of Hartree-Fock theory in which the more
computationally expensive integrals are replaced by adjustable
parameters, which are determined by fitting experimental atomic and
molecular data. Different choices of parameterization lead to
different specific theories (e.g., MNDO, AM1, PM3). Semiempirical
calculations are much faster than ab initio calculations.

size-consistent

Describes a calculation that gives the same energy for two atoms (or
molecular fragments) separated by a large distance as is obtained from
summing the energies for the atoms (or molecular fragments) computed separately.
So for a size-consistent method, the bond energy in N2 is De
= 2E(N) - E(N2). For a method that is not size-consistent, a
"supermolecule" calculation with a big distance
(e.g., 100 ┼) is required:
De = E(N......N) - E(N2).

Refers to a basis set that is more than minimal (see MBS)
for the valence orbitals, i.e., at least VDZ. 3-21G
is one example of a split-valence basis. 6-311G might
be called a triple-split-valence basis.

Slater-type orbital. Basis function with an exponential radial function,
i.e., exp(- zeta r). Also used to denote a fit to such a function using
other functions, such as gaussians. For example, STO-3G
is an MBS that uses 3 gaussians to fit an exponential.
Exponentials are probably better basis functions than gaussians, but are
so much more difficult computationally that they were abandoned by most
people a long time ago.

An indication of how far the usual HF orbitals
differ from the Brueckner orbitals. It has been used as an indicator of
multi-reference character and therefore of the reliability of coupled cluster
calculations (18,19), although this
usage has been challenged.

Spin-unrestricted Hartree-Fock.
For open-shell molecules. There are
separate orbitals for spin-up (alpha) and for spin-down (beta) electrons.
UHF
wavefunctions are usually not eigenfunctions of spin, and are often contaminated
by states of higher spin multiplicity (which usually raises the energy;
see PUHF).

Literature Cited

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